I sew on a 401 regularly. I also have a 503 (same as 403 in 60's looking body, nicknamed the Rockateer). They are essentially the same except that the 403/503 has removable cams. The 401 and 501 have built-in cams. It's often said that the 401 is the best sewing machine Singer ever made. It has really close tolerances, so it may make take extra effort on the cleaning/removing old oil from the moveable parts. I've used 90% alcohol from Walmart and unscented lamp soil, also from Walmart to remove the old sticky residue. I've used q-tips, strips of old T-shirts and toothpicks saturated in one of the above solvents. Of course, mine was totally frozen up when I bought it. Yours may not take that much work.
There is a series on youtube about the 401. Here's a link:
Link to youtube.
bkay